Linear functional form

\[Y = \beta_0 + \beta_{1}X\]

Linear functional form

Least squares regression

Linear functional form

## 
## Call:
## lm(formula = y ~ x, data = sim1)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -4.147 -1.520  0.133  1.467  4.652 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    4.221      0.869    4.86  4.1e-05 ***
## x              2.052      0.140   14.65  1.2e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.2 on 28 degrees of freedom
## Multiple R-squared:  0.885,  Adjusted R-squared:  0.88 
## F-statistic:  215 on 1 and 28 DF,  p-value: 1.17e-14

Linear functional form

Residuals and visualizations

Residuals and visualizations

Residuals and visualizations

Non-linearity of the data

Outliers

High-leverage points

\[\text{Influence} = \text{Leverage} \times \text{Discrepancy}\]

• Leverage
• Discrepancy

High-leverage points

• Leverage statistic/hat value

  \[h_i = \frac{1}{n} + \frac{(x_i - \bar{x})^2}{\sum_{i'=1}^{n} (x_{i'} - \bar{x})^2}\]

• Residuals

• Cook’s D

High-leverage points

• Outcome of interest - number of federal laws struck down by SCOTUS

1. Age - the mean age of the members of the Supreme Court
2. Tenure - mean tenure of the members of the Court
3. Unified - a dummy variable indicating whether or not the Congress was controlled by the same party in that period

Happiness and gender

Happiness and gender

##    Cell Contents 
## |-------------------------|
## |                   Count | 
## |           Total Percent | 
## |-------------------------|
## 
## =======================================
##                  happy$sex
## happy$happy       male   female   Total
## ---------------------------------------
## not too happy    1904     2460    4364 
##                   5.5%     7.1%        
## ---------------------------------------
## pretty happy     8760    10539   19299 
##                  25.2%    30.3%        
## ---------------------------------------
## very happy       4833     6327   11160 
##                  13.9%    18.2%        
## ---------------------------------------
## Total           15497    19326   34823 
## =======================================

Happiness and gender

##    Cell Contents 
## |-------------------------|
## |                   Count | 
## |             Row Percent | 
## |          Column Percent | 
## |-------------------------|
## 
## =======================================
##                  happy$sex
## happy$happy       male   female   Total
## ---------------------------------------
## not too happy    1904     2460    4364 
##                  43.6%    56.4%   12.5%
##                  12.3%    12.7%        
## ---------------------------------------
## pretty happy     8760    10539   19299 
##                  45.4%    54.6%   55.4%
##                  56.5%    54.5%        
## ---------------------------------------
## very happy       4833     6327   11160 
##                  43.3%    56.7%   32.0%
##                  31.2%    32.7%        
## ---------------------------------------
## Total           15497    19326   34823 
##                  44.5%    55.5%        
## =======================================

Mosaic plot

Mosaic plot

Proportional bar chart

Dot plot for summary statistics

Generalized linear models

Odds of survival